a geometry teacher has a set of 60 plastic pentagons and octagons. she happened to notice that all the figures together have a total of 354 sides. how many of each shape are there.

Question

GPT-4o mini · Answer

Let $ p $ be the number of pentagons and $ o $ be the number of octagons. We can set up the following system of equations based on the information provided: 1. The total number of shapes is 60: $$ p + o = 60 $$ 2. The total number of sides is 354: $$ 5p + 8o = 354 $$ We can solve this system of equations step by step. First, from the first equation, we can express $ o $ in terms of $ p $: $$ o = 60 - p $$ Next, we substitute this expression for $ o $ into the second equation: $$ 5p + 8(60 - p) = 354 $$ Expanding this gives: $$ 5p + 480 - 8p = 354 $$ Combining like terms, we have: $$ -3p + 480 = 354 $$ Subtracting 480 from both sides: $$ -3p = 354 - 480 $$ $$ -3p = -126 $$ Dividing by -3: $$ p = 42 $$ Now that we have $ p $, we can find $ o $ using the expression we found earlier: $$ o = 60 - p = 60 - 42 = 18 $$ Thus, the teacher has $ \boxed{42} $ pentagons and $ \boxed{18} $ octagons. Let's verify our solution: - The total number of shapes is $ 42 + 18 = 60 $ (correct). - The total number of sides is $ 5(42) + 8(18) = 210 + 144 = 354 $ (correct). Everything checks out!